Definition 111.24.1. Let $\phi : A \to B$ be a homomorphism of rings. We say that the *going-up theorem* holds for $\phi $ if the following condition is satisfied:

for any ${\mathfrak p}, {\mathfrak p}' \in \mathop{\mathrm{Spec}}(A)$ such that ${\mathfrak p} \subset {\mathfrak p}'$, and for any $P \in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}$, there exists $P'\in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}'$ such that $P \subset P'$.

Similarly, we say that the *going-down theorem* holds for $\phi $ if the following condition is satisfied:

for any ${\mathfrak p}, {\mathfrak p}' \in \mathop{\mathrm{Spec}}(A)$ such that ${\mathfrak p} \subset {\mathfrak p}'$, and for any $P' \in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}'$, there exists $P\in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}$ such that $P \subset P'$.

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